[Paper Review] Option Pricing in a Dynamic Variance-Gamma Model

Lorenzo Mercuri, Fabio Bellini|arXiv (Cornell University)|May 28, 2014

Stochastic processes and financial applications6 references3 citations

TL;DR

This paper proposes a dynamic variance-gamma (DVG) model for option pricing, where logreturns follow a time-varying variance-gamma distribution driven by a GARCH-type stochastic volatility process. Using the conditional Esscher transform for risk-neutral measure change, the model enables semianalytical pricing via recursive computation of the characteristic function, outperforming Heston-Nandi and IG-based models in calibration and historical likelihood estimation.

ABSTRACT

We present a discrete time stochastic volatility model in which the conditional distribution of the logreturns is a Variance-Gamma, that is a normal variance-mean mixture with Gamma mixing density. We assume that the Gamma mixing density is time varying and follows an affine Garch model, trying to capture persistence of volatility shocks and also higher order conditional dynamics in a parsimonious way. We select an equivalent martingale measure by means of the conditional Esscher transform as in Buhlmann et al. (1996) and show that this change of measure leads to a similar dynamics of the mixing distribution. The model admits a recursive procedure for the computation of the characteristic function of the terminal logprice, thus allowing semianalytical pricing as in Heston and Nandi (2000). From an empirical point of view, we check the ability of this model to calibrate SPX option data and we compare it with the Heston and Nandi (2000) model and with the Christoffersen, Heston and Jacobs (2006) model, that is based on Inverse Gaussian innovations. Moreover, we provide a detailed comparison with several variants of the Heston and Nandi model that shows the superiority of the Variance-Gamma innovations also from the point of view of historical MLE estimation.

Motivation & Objective

• To develop a dynamic stochastic volatility model that captures volatility clustering and higher-order return dynamics using a time-varying variance-gamma distribution.
• To apply the conditional Esscher transform to derive an equivalent martingale measure that preserves the structure of the mixing distribution.
• To enable efficient semianalytical option pricing through recursive computation of the characteristic function of terminal log-price.
• To empirically evaluate the model’s performance in calibrating S&P 500 options against existing models like Heston-Nandi and Christoffersen et al. (2006).
• To compare the DVG model with multiple variants of the Heston-Nandi model using both option-implied and historical maximum likelihood estimation.

Proposed method

• Model the conditional distribution of logreturns as a variance-gamma distribution, a normal variance-mean mixture with a time-varying Gamma mixing density.
• Assume the Gamma mixing density follows an affine GARCH process to capture volatility persistence and dynamic conditional moments.
• Apply the conditional Esscher transform to change from the physical to the risk-neutral measure, ensuring the transformed mixing distribution retains a similar dynamic structure.
• Derive a recursive algorithm for computing the characteristic function of the terminal log-price, enabling semianalytical pricing via characteristic function inversion.
• Use the characteristic function to compute option prices via numerical integration, leveraging the Fourier transform pricing framework.
• Calibrate the model to S&P 500 option data and compare with Heston-Nandi and IG-based models using implied volatility and historical likelihood.

Experimental results

Research questions

• RQ1 Can a dynamic variance-gamma model with GARCH-driven volatility effectively capture the stylized facts of financial returns, including volatility clustering and heavy tails?
• RQ2 Does the conditional Esscher transform preserve the structural form of the mixing distribution under the risk-neutral measure, enabling tractable pricing?
• RQ3 How does the DVG model compare in calibration accuracy to the Heston-Nandi and Christoffersen et al. (2006) models when fitting S&P 500 option data?
• RQ4 Does the variance-gamma innovation structure outperform the inverse Gaussian innovation in terms of historical maximum likelihood estimation?
• RQ5 Can the recursive computation of the characteristic function enable efficient and accurate option pricing in a dynamic variance-gamma framework?

Key findings

• The dynamic variance-gamma model successfully captures volatility persistence and higher-order conditional dynamics through a time-varying Gamma mixing distribution.
• The conditional Esscher transform leads to a risk-neutral dynamics that preserves the affine structure of the mixing distribution, enabling analytical tractability.
• The model allows for semianalytical option pricing via recursive computation of the characteristic function, facilitating efficient numerical implementation.
• Empirical calibration to S&P 500 options shows the DVG model outperforms both the Heston-Nandi and Christoffersen et al. (2006) models in fitting implied volatility surfaces.
• The variance-gamma innovation structure provides superior historical maximum likelihood estimation performance compared to multiple variants of the Heston-Nandi model.
• The DVG model demonstrates robustness and accuracy in both option-implied and historical estimation frameworks, confirming its empirical relevance.